Quadratic equations are guaranteed on every GCSE Higher and AS-Level Maths paper. You need to solve by factorising, by completing the square, and by using the formula — and to use the discriminant to determine the number of real roots.
A-Level builds on these methods for curve sketching, simultaneous equations and quadratic inequalities. Strong algebra here pays off across mechanics, statistics and pure modules later.
At GCSE Higher you solve quadratics by factorising, completing the square and using the quadratic formula, and sketch parabolas showing roots and the turning point. Word problems leading to a quadratic (areas, projectile heights) are routine.
At A-Level you use the discriminant to determine the number of real roots, solve quadratic inequalities, and apply quadratics to simultaneous equations with a line and a curve. Completing the square is a key tool for proofs, integration setups and graph transformations.
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Q: Write down the quadratic formula.
A: x = (−b ± √(b² − 4ac)) / 2a, for ax² + bx + c = 0.
Q: What does the discriminant tell you?
A: Δ = b² − 4ac. Positive: two real roots; zero: one repeated root; negative: no real roots.
Q: Solve x² − 6x + 8 = 0 by factorising.
A: (x − 2)(x − 4) = 0 → x = 2 or x = 4.
Q: Complete the square for x² + 6x + 4.
A: (x + 3)² − 5.
Use factorising first if the equation has nice integer roots. If it doesn't factorise neatly, use the formula or complete the square.
Once a quadratic is in the form a(x − h)² + k, the minimum (or maximum if a < 0) is at (h, k) — you can read the vertex straight off.
The parabola does not cross the x-axis, so there are no real solutions. (At Further Maths there are two complex conjugate roots.)
Substitute each root back into the original equation — both sides should equal zero.
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